Plasmonic graphene devices

ABSTRACT

An integrated graphene-based structure comprises an N-dimensional array of elements formed on a surface of a substrate. The N-dimensional array of elements includes a plurality of rows. Each respective row in the plurality of rows comprises a corresponding plurality of elements formed along a first dimension. Each element in the corresponding plurality of elements comprising at least one graphene stack and separated from an adjacent element along the first dimension by a first average spatial separation thereby resulting in a first periodicity in lateral spacing along the first dimension. Each respective row in the plurality of rows is separated from an adjacent row along a second dimension by a second average spatial separation, thereby resulting in a second periodicity in lateral spacing along the second dimension. The N-dimensional array exhibits a set of characteristic electromagnetic interference properties in response to electromagnetic radiation incident on the N-dimensional array.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Patent Application No.61/802,006, entitled “Plasmonic Device Enhancements,” filed Mar. 15,2013, which is hereby incorporated by reference herein in its entirety.

TECHNICAL FIELD

The disclosed embodiments relate generally to devices and methods offabricating devices with graphene-based structures and topographies andmore specifically to plasmonic and electromagnetic interferenceproperties of graphene-based structures and topographies.

BACKGROUND

Electromagnetic interference properties of plasmonic arrays andspatially periodic structures that demonstrate plasmonic resonanceeffects are of significant research and commercial interest owing totheir ease of use and deployment in optical, mechanical,electromechanical systems and the like.

Furthermore, graphene-based structures (such as graphene quantum dots,graphene nanoribbons, graphene nanonetworks, graphene plasmonics, andgraphene super-lattices) exhibit superior chemical, mechanical,electronic, and optical properties that have applications and benefitsin various electronic devices, composite materials, and implementationsfor energy generation and storage.

Accordingly, there is a need in the art for improved devices withplasmonic properties and electromagnetic resonance properties thatleverage the improved optical, electrical and mechanical properties ofgraphene-based structures to enhance plasmonic interactions betweenelectromagnetic radiation and graphene.

SUMMARY

This present disclosure addresses the above-identified need and providesimproved devices with plasmonic properties and electromagnetic resonanceproperties. Such graphene-based plasmonic structures lend themselves touse in improved optical and electronic and mechanical devices, systemsand components (such as optical lenses, optical filters, DNA sequencingmicrocavities, a DNA sequencing nanocavities, condensor lenses, adiffraction devices for optical absorption and the like.

In particular, disclosed are graphene-based topographies and devices andmethods of fabricating the devices that enable the use of graphene-basedstructures for improved optical and electromagnetic properties. Anintegrated graphene-based structure comprises an N-dimensional array ofelements formed on a surface of a substrate. The N-dimensional array ofelements includes a plurality of rows. Each respective row in theplurality of rows comprises a corresponding plurality of elements formedalong a first dimension, the first dimension characterized by an axis ofthe respective row, each element in the corresponding plurality ofelements (a) comprising at least one graphene stack and (b) separatedfrom an adjacent element along the first dimension by a first averagespatial separation thereby resulting in a first periodicity in lateralspacing along the first dimension. Each respective row in the pluralityof rows is separated from an adjacent row along a second dimension by asecond average spatial separation, thereby resulting in a secondperiodicity in lateral spacing along the second dimension. TheN-dimensional array exhibits a set of characteristic electromagneticinterference properties in response to electromagnetic radiationincident on the N-dimensional array.

In some embodiments, the set of characteristic electromagneticinterference properties is associated with an interaction of the atleast one respective graphene stack with the incident electromagneticradiation. In some embodiments, the set of characteristicelectromagnetic interference properties comprises a first plasmonicresonance, occurring at a first resonance wavelength of incidentelectromagnetic radiation, characterized by the first periodicity inlateral spacing along the first dimension and determined in accordancewith the first average spatial separation along the first dimension.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed asprojections from a substrate with one or more graphene stacks on the topsurfaces of the elements distal from the native surface of thesubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 1B illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 a-1 b of FIG. 1A and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 1C illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of elements spacednon-uniformly and formed as trenches engraved into a substrate with oneor more graphene stacks on the trench surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 1D illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 c-1 d of FIG. 1C and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 1E illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe native surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure.

FIG. 1F illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 e-1 f of FIG. 1E and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 1G illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising a substantially radial arrangementof rows of elements formed as trenches engraved into a substrate withone or more graphene stacks on the native surface of the substrate andone or more graphene stacks in the trench surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 1H illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 g-1 h of FIG. 1G and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 1I illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed asprojections from a substrate with one or more graphene stacks on thebottom surfaces of the elements proximal to the native surface of thesubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 1J illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 i-1 j of FIG. 1I and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 1K illustrates a top view of an exemplary graphene devicetopography (e.g., a plurality of elements along a row of an array formedas trenches engraved into a substrate with one or more graphene stackson the side walls of the engraved trenches) fabricated by exemplarymethods in accordance with an embodiment of the present disclosure

FIG. 1L illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 k-1 l of FIG. 1K and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 2A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of non-uniformlyspaced elements formed as projections from a substrate with one or moregraphene stacks on the top surfaces of the elements distal from thenative surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure.

FIG. 2B illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 2 a-2 b of FIG. 2A and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 2C illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe trench surfaces of the trenches and a secondary layer formed overthe one or more graphene stacks of one or more of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 2D illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 2 c-2 d of FIG. 2C and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 2E illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe native surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure.

FIG. 2F illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 2 e-2 f of FIG. 2E and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIGS. 3A-3C illustrate top views of exemplary graphene devicetopographies fabricated by exemplary methods in accordance withembodiments of the present disclosure.

FIG. 3D illustrates a cross-sectional view of exemplary graphene devicetopography (e.g., a row of elements with one or more graphene stackscovered with one or more secondary materials formed as projections on asubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 4A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of non-uniformlyspaced elements engraved as one or more layers of trenches into thesubstrate with one or more graphene stacks on the trench surfaces of theelements) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 4B illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 4 a-4 b of FIG. 4A and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 5A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising hexagonal sub-arrayselements formed as projections on the substrate with one or moregraphene stacks on respective top or bottom surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 5B and 5C illustrate cross-sectional views of the exemplarygraphene device topography taken along lines 5 b-5 b′ and 5 c-5 c′ ofFIG. 5A and fabricated by an exemplary method in accordance with anembodiment of the present disclosure.

FIG. 6 illustrates a cross-sectional view of exemplary graphene devicetopography (e.g., quantum dots covered with one or more graphene stacks)fabricated by an exemplary method in accordance with an embodiment ofthe present disclosure.

FIGS. 7A-7H illustrate cross-sectional views of graphene devicetopography (e.g., including a plurality of contiguous cavities engravedinto the substrate) fabricated by an exemplary method in accordance withan embodiment of the present disclosure.

FIGS. 8A, 8C and 8E illustrate top views of an exemplary graphene devicetopography (e.g., elements along a row of an array formed as projectionswith shapes corresponding to frustums of pyramids on the substrate withone or more graphene stacks on respective top surfaces, side walls, anda combination thereof) fabricated by exemplary methods in accordancewith an embodiment of the present disclosure.

FIGS. 8B, 8D and 8F illustrate cross-sectional views of the exemplarygraphene device topography taken along lines 8 a-8 b, 8 c-8 d and 8 e-8f of FIGS. 8A, 8C and 8E and fabricated by an exemplary method inaccordance with an embodiment of the present disclosure.

FIGS. 9A-9E illustrate a flow diagram representing exemplary processflow for the fabrication of one or more graphene device topography inaccordance with an embodiment of the present disclosure.

Like reference numerals refer to corresponding parts throughout thedrawings.

DESCRIPTION OF EMBODIMENTS

It will be understood that, although the terms “first,” “second,” etc.may be used herein to describe various elements, these elements shouldnot be limited by these terms. These terms are only used to distinguishone element from another. For example, a first element could be termed asecond element, and, similarly, a second element could be termed a firstelement, without changing the meaning of the description, so long as alloccurrences of the “first element” are renamed consistently and alloccurrences of the second element are renamed consistently. The firstelement and the second element are both elements, but they are not thesame element.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the claims. Asused in the description of the embodiments and the appended claims, thesingular forms “a”, “an” and “the” are intended to include the pluralforms as well, unless the context clearly indicates otherwise. It willalso be understood that the term “and/or” as used herein refers to andencompasses any and all possible combinations of one or more of theassociated listed items. It will be further understood that the terms“comprises” and/or “comprising,” when used in this specification,specify the presence of stated features, integers, steps, operations,elements, and/or components, but do not preclude the presence oraddition of one or more other features, integers, steps, operations,elements, components, and/or groups thereof.

As used herein, the term “if” may be construed to mean “when” or “upon”or “in response to determining” or “in accordance with a determination”or “in response to detecting,” that a stated condition precedent istrue, depending on the context. Similarly, the phrase “if it isdetermined [that a stated condition precedent is true]” or “if [a statedcondition precedent is true]” or “when [a stated condition precedent istrue]” may be construed to mean “upon determining” or “in response todetermining” or “in accordance with a determination” or “upon detecting”or “in response to detecting” that the stated condition precedent istrue, depending on the context.

Reference will now be made in detail to various embodiments, examples ofwhich are illustrated in the accompanying drawings. In the followingdetailed description, numerous specific details are set forth in orderto provide a thorough understanding of the invention and the describedembodiments. However, the invention may be practiced without thesespecific details. In other instances, well-known methods, procedures,components, and circuits have not been described in detail so as not tounnecessarily obscure aspects of the embodiments.

In some embodiments, graphite-based structures, e.g. graphene quantumdots, graphene nanoribbons (GNRs), graphene nanonetworks, grapheneplasmonics and graphene super-lattices, exhibit many exceptionalchemical, mechanical, electronic and optical properties, and are verydesirable for use in electronic devices, composite materials, and energygeneration and storage. Such graphite-based structures in generalcomprise a graphene layer, typically nanometers thick and having acharacteristic dimension also in the nanometers range. For example, inorder to obtain adequate band gaps for operation at room temperature,GNRs are required to have a width within a few nanometers due to theinverse relationship between the band gap and the width of the GNRs.

In some embodiments, various methods are provided for fabricatinggraphite-based structures while achieving desired size, specifiedgeometries, and characterized electronic properties of thegraphite-based structures. These methods include, but are not limitedto, (1) the combination of e-beam lithography and oxygen plasma etching;(2) stripping of graphite that is sonochemically processed; and (3)bottom-up chemical synthesis, e.g., by cyclodehydrogenation of1,4-diiodo-2,3,5,6-tetraphenylbenzene6, or10,10′-dibromo-9,9′-bianthryl, polyanthrylene oligomers self-assembledon Au(111), Ag(111) or silica substrates, to name a few examples.

In some embodiments, different pitch and duty cycle combinations ingraphene devices are utilized to improve efficiency. In particular, insome embodiments, graphene sheets are stacked, with different pitch andcritical dimensions, such that devices have multiple pass functionality.Similarly, in some embodiments, structures comprising multiple levels ofgraphene layers allow for more versatile and efficient band gap devices.

Embodiments of the present disclosure are described in the context ofmethods for fabricating thin films from layered materials and in thecontext of thin films made therefrom. In this specification and claims,layered materials refer to a material comprising a plurality of sheets,with each sheet having a substantially planar structure.

As used herein, the term “thin films” refers to a thin layer comprisingone sheet (e.g, a sheet of graphene); it also refers to several, severaltens, hundreds or thousands of such sheets. The thickness of the thinfilms can range from a nanometer to several micrometers, or to severaltens of micrometers. Final thin films produced by some processesdisclosed in this application have a thickness in nanometers, andpreferably less than fifty nanometers. Similarly, as used herein, a“graphene layer” refers to several, several tens, several hundreds orseveral thousands of such sheets. As user herein a sheet is a sheet ofgraphene, which is a single sheet composed of sp²-hybridized carbon.

As used herein, the term “stacks” refers to one or more layers of amaterial (e.g., one or more layers of graphene). Like “thin films,”“stacks” can also refer to several, several tens, several hundreds orseveral thousands of layers of material. For example, a stack ofgraphene refers to one or more layers of graphene or graphenestructures. As used herein, the term “graphene structures” is usedinterchangeably with “graphene.” As used herein, the term “stacks” isinterchangeable with the terms “graphene stacks” and “stacks ofgraphene.”

As used herein, the terms “graphene based nanostructure” and “graphenenanostructure” are interchangeable and refer to any carbon basedstructure incorporating graphene. Examples of graphene basednanostructures include, but are not limited to, graphene nanoribbons,graphene nanonetworks, graphene poles/pillars, and graphene basednanohole superlattices.

As used herein, the term “level” refers to one or more graphene stacksfor a given foundation layer or substrate. Thus, in some embodiments, alevel of graphene contains multiple graphene stacks formed from arespective foundation layer or substrate. As sometimes used herein,“level” is shorthand for “graphene level” or “level of graphene.”

As used herein, the term “substrate” refers to one layer or multiplelayers. In some embodiments, a substrate is glass, Si, SiO₂, SiC, oranother material. When referring to multiple layers, the term“substrate” is equivalent to and interchangeable with the term“substrate stack.”

As used herein, the term “foundation material” refers to any materialthat is suitable for growing graphene. In some embodiments, foundationmaterials are catalytic metals, e.g., Pt, Au, Fe, Rh, Ti, Ir, Ru, Ni, orCu. In some other embodiments, foundation materials are non-metalmaterials, such as Si, SiC, non-stoichiometric SiC (e.g., boron doped orotherwise), and other carbon enhanced materials. As used herein, thephrase “carbon enhanced” materials refers to any materials into whichcarbon has been added.

In some implementations, the substrate used in the present disclosure isglass, silicon, SiC, SiO₂, or SiC/Si. In some embodiments, the substrateis a solid substance in a form of a thin slice. In some embodiments, thesubstrate is planar. In some embodiments the substrate is flexible. Insome embodiments the substrate is rigid. In various embodiments, thesubstrate is made of a dielectric material, a semiconducting material, ametallic material, or a combination of such materials. Exemplarydielectric materials include glass, silicon dioxide, neoceram, andsapphire. Exemplary semiconducting materials include silicon (Si),silicon carbide (SiC), germanium (Ge), boron nitride (BN), andmolybdenum sulfide (MoS). Exemplary metallic materials comprise copper(Cu), nickel (Ni), platinum (Pt), gold (Au), cobalt (Co), ruthenium(Ru), palladium (Pd), titanium (Ti), silver (Ag), aluminum (Al), cadmium(Cd), iridium (Ir), combinations thereof, and alloys thereof. In someembodiments the substrate comprises Si, SiO₂, SiC, Cu, Ni, or othermaterials. In some embodiments, the substrate substantially comprisesneoceram, borosilicate glass, germanium arsenide, a IV-V semiconductormaterial, a substantially metallic material, a high temperature glass,or a combination thereof.

In some embodiments, the substrate substantially comprises SiO₂ glass,soda lime glass, lead glass, doped SiO₂, aluminosilicate glass,borosilicate glass, dichroic glass, germanium/semiconductor glass, glassceramic, silicate/fused silica, soda lime glass, quartz orchalcogenide/sulphide glass, fluoride glass, a glass-based phenolic,flint glass, or cereated glass.

In some embodiments, the substrate is made of poly methyl methacrylate(PMMA), polyethylene terephthalate (PET), polyvinyl alcohol (PVA), orcellulose acetate (CA). In some embodiments, the substrate is made of aurethane polymer, an acrylic polymer, a fluoropolymer,polybenzamidazole, polymide, polytetrafluoroethylene,polyetheretherketone, polyamide-imide, glass-based phenolic,polystyrene, cross-linked polystyrene, polyester, polycarbonate,polyethylene, polyethylene, acrylonitrile-butadiene-styrene,polytetrafluoro-ethylene, polymethacrylate, nylon 6,6, cellulose acetatebutyrate, cellulose acetate, rigid vinyl, plasticized vinyl, orpolypropylene.

In some embodiments, the substrate includes one layer. In alternativeembodiments, the substrate includes a plurality of layers. In someembodiments, a substrate comprises a plurality of layers, each with adifferent material. In some embodiments, a layer of another substance isapplied onto the substrate. In some embodiments, the substrate hascrystallographic symmetry.

FIG. 1A illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed asprojections from a substrate with one or more graphene stacks on the topsurfaces of the elements distal from the native surface of thesubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 1B illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 a-1 b of FIG. 1A and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

As shown in FIG. 1A, an integrated graphene-based structure comprises anN-dimensional array of elements formed on a surface of a substrate. TheN-dimensional array of elements includes a plurality of rows. Eachrespective row in the plurality of rows comprises a correspondingplurality of elements formed along a first dimension, the firstdimension characterized by an axis of the respective row, each elementin the corresponding plurality of elements (a) comprising at least onegraphene stack and (b) separated from an adjacent element along thefirst dimension by a first average spatial separation thereby resultingin a first periodicity in lateral spacing along the first dimension.Each respective row in the plurality of rows is separated from anadjacent row along a second dimension by a second average spatialseparation, thereby resulting in a second periodicity in lateral spacingalong the second dimension. The N-dimensional array exhibits a set ofcharacteristic electromagnetic interference properties in response toelectromagnetic radiation incident on the N-dimensional array.

In some embodiments, the set of characteristic electromagneticinterference properties is associated with an interaction of the atleast one respective graphene stack with the incident electromagneticradiation. In some embodiments, the set of characteristicelectromagnetic interference properties comprises a first plasmonicresonance, occurring at a first resonance wavelength of incidentelectromagnetic radiation, characterized by the first periodicity inlateral spacing along the first dimension and determined in accordancewith the first average spatial separation along the first dimension.

In some embodiments, the set of characteristic electromagneticinterference properties comprises a second plasmonic resonance,occurring at a second resonance wavelength of incident electromagneticradiation, characterized by the second periodicity in lateral spacingalong the second dimension and determined in accordance with the secondaverage spatial separation along the second dimension.

In some embodiments, the set of characteristic electromagneticinterference properties is associated with an interaction of a portionof each of one or more of the elements in the N-dimensional array withthe incident electromagnetic radiation. In some embodiments, therespective portion of an element in the one or more of the elements inthe N-dimensional array comprises a corresponding graphene stack; andthe set of characteristic electromagnetic interference properties isassociated with an interaction of the corresponding graphene stack withthe incident electromagnetic radiation.

In some embodiments, the first average spatial separation issubstantially equal to the second average spatial separation. In someembodiments, as shown in FIGS. 1A-1B a first row in the plurality ofrows comprises a first element and a second element. The first elementhas a first height and a first width. The second element has a secondheight and a second width and a reference position of the first elementis spaced at a first distance from a reference position of the secondelement. In some embodiments, the first height is different than thesecond height.

In some embodiments, a difference between the first height and thesecond height is between 5 nanometers and 1000 nanometers. In someembodiments, the first width is different than the second width. In someembodiments, the first distance has a value between 50 nanometers and200 nanometers. In some embodiments, a graphene stack in the at leastone graphene stack comprises between 1 sheet and 500 sheets of graphene.In some embodiments, a graphene stack in the at least one graphene stackhas a thickness between 0.3 nanometers and 150 nanometers. In someembodiments, a graphene stack in the at least one graphene stack has awidth between 3 nanometers and 100 nanometers.

In some embodiments, the first average spatial separation has a valuebetween 2 nanometers and 150 nanometers. In some embodiments, the secondaverage spatial separation has a value between 2 nanometers and 150nanometers. In some embodiments, N is a positive integer of value two orgreater. In some embodiments, the second dimension is substantiallyorthogonal to the first dimension.

As shown in FIG. 1B, in some embodiments, the projection includes afirst end proximal to the native surface of the substrate and a secondend distal to the surface of the substrate and a graphene stack in theat least one graphene stack is formed on the second end of therespective projection.

In some embodiments, a secondary layer is formed on the first end of theprojection and the secondary layer comprises a secondary material. Insome embodiments, the secondary layer is a passivation layer and thesecondary material is a transmissive material. In some embodiments, thesecondary layer is a waveguide tuned to a predetermined wavelength. Insome embodiments, the secondary material comprises silicon nitride,silicon dioxide, titanium dioxide, tantalum oxide, hafnium oxide, or anycombination thereof.

In some embodiments, the secondary layer is an electrically isolatinglayer and the secondary material comprises a dielectric. In someembodiments, the secondary material comprises a transparent conductiveoxide. In some embodiments, the secondary material comprises anelectrically conductive material and the secondary layer is aninterconnect lead to electrically connect a first conductive materialwith a second conductive material. In some embodiments, the secondarymaterial comprises an electrically conductive material and the secondarylayer redirects energy collected by the graphene stack. In someembodiments, the secondary material comprises a metal. In someembodiments, the secondary material comprises aluminum, platinum,copper, nickel, palladium, tungsten, or any combination thereof. In someembodiments, the secondary layer is a semiconductor device and thesecondary material comprises one or more semiconductors, one or moredoped semiconductors, or a combination thereof. In some embodiments, thesemiconductor device comprises a diode, a resistor, a schottky junction,or any combination thereof. In some embodiments, the secondary materialcomprises a dielectric material and the secondary layer is a componentof a capacitor or a charge collection device or a combination thereof.In some embodiments, the secondary layer is a component of a capacitoror a charge collection device or a combination thereof. In someembodiments, the N-dimensional array of elements forms a component of asensor, a detector, an antenna, an amplification device, a responseanalyzer for one or more plasmonic reactions, or any combinationthereof. In some embodiments, the secondary layer forms a component ofthe sensor, the detector, the antenna, the amplification device, or theresponse analyzer for one or more plasmonic reactions.

In some embodiments, the plurality of elements includes a first elementcomprising a first graphene stack and a second element comprising asecond graphene stack, the first element is characterized by a firstwidth and the second element is characterized by a second width, wherethe first width is different from the second width, and a referenceposition of the first element is separated from a reference position ofthe second element by a first spatial separation.

In some embodiments, the set of characteristic electromagneticinterference properties comprises: a first elemental resonance,occurring at a first elemental wavelength of incident electromagneticradiation, determined in accordance with the first lateral width of thefirst element and associated with an interaction of the first graphenestack with the incident electromagnetic radiation; a second elementalresonance, occurring at a second elemental wavelength of incidentelectromagnetic radiation, determined in accordance with the secondlateral width of the second element and associated with an interactionof the second graphene stack with the incident electromagneticradiation; and, optionally, a respective inter-elemental resonance,occurring at a third elemental wavelength of incident electromagneticradiation, determined in accordance with the first spatial separation.

FIG. 1C illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of elements spacednon-uniformly and formed as trenches engraved into a substrate with oneor more graphene stacks on the trench surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 1D illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 c-1 d of FIG. 1C and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

In some embodiments, each respective element in the correspondingplurality of elements comprises a corresponding cavity in a plurality ofcavities engraved into the substrate in a direction substantiallyorthogonal to the native surface of the substrate.

In some embodiments, the corresponding cavity is bounded by a first endproximal to the native surface of the substrate and a second end distalto the native surface of the substrate and a first graphene stack in theat least one graphene stack is formed at the second end of therespective cavity.

In some embodiments, the plurality of rows comprises a first row and asecond row and the corresponding plurality of elements in the first rowis offset relative to the corresponding plurality of elements in thesecond row by an average alignment distance.

In some embodiments, the respective plurality of elements of arespective row in the plurality of rows includes a first element, asecond element adjacent to the first element, and a third elementadjacent to the second element in the respective row; a referenceposition of the first element is at a first distance from a referenceposition of the second element; the reference position of the secondelement is a second distance from a reference position of the thirdelement; and the first distance is different than the second distance,thereby resulting in a non-linear array of elements.

In some embodiments, the plurality of rows includes a first row, asecond row adjacent to the first row, and a third row adjacent to thesecond row; a reference axis of the first row is spaced at a firstdistance from a reference axis of the second row; the reference axis ofthe second row is spaced at a second distance from a reference axis ofthe third row; and the first distance is different than the seconddistance, thereby resulting in a non-linear array of elements. In someembodiments, the first and second distance is respectively a first andsecond predefined linear separation of values between 2 nanometers and150 nanometers.

In some embodiments, the corresponding plurality of elements includes afirst element comprising a first cavity characterized by a first width,a second element adjacent to the first element comprising a secondcavity characterized by a second width, and a third element adjacent tothe second element comprising a third cavity characterized by a thirdwidth; a reference position of the first element is a first distancefrom a reference position of the second element; and the referenceposition of the second element is a second distance from a referenceposition of the third element.

In some embodiments, the first distance is different than the seconddistance. In some embodiments, the first distance is equal to the seconddistance. In some embodiments, the first width is equal to the secondwidth and the second width is substantially equal to the third width. Insome embodiments, the first width is less than the second width and thesecond width is less than the third width; and the difference betweenthe second width and the first width is equal to a difference betweenthe third width and the second width. In some embodiments, the firstwidth is between 1 nanometer and 300 nanometers, the second width isbetween 1 nanometer and 300 nanometers, and the third width is between 1nanometer and 300 nanometers.

In some embodiments, the plurality of elements includes a first elementcomprising a first graphene stack and a second element comprising asecond graphene stack; the first element is characterized by a firstwidth and the second element is characterized by a second width, wherethe first width is different from the second width; and a referenceposition of the first element is separated from a reference position ofthe second element by a first spatial separation.

In some embodiments, the set of characteristic electromagneticinterference properties comprises: a first elemental resonance,occurring at a first elemental wavelength of incident electromagneticradiation, determined in accordance with the first lateral width of thefirst element and associated with an interaction of the first graphenestack with the incident electromagnetic radiation; a second elementalresonance, occurring at a second elemental wavelength of incidentelectromagnetic radiation, determined in accordance with the secondlateral width of the second element and associated with an interactionof the second graphene stack with the incident electromagneticradiation; and, optionally, a respective inter-elemental resonance,occurring at a third elemental wavelength of incident electromagneticradiation, determined in accordance with the first spatial separation.

In some embodiments, the first element comprises a first cavity in thesubstrate in a direction substantially orthogonal to the surface of thesubstrate and the second element comprises a second cavity in thesubstrate in the direction substantially orthogonal to the surface ofthe substrate. In some embodiments, the first cavity includes a firstend proximal to the native surface of the substrate and a second enddistal to the native surface of the substrate; and the first graphenestack is formed on the second end of the first cavity.

FIG. 1E illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe native surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure.

FIG. 1F illustrates a cross-sectional view of the exemplary graphenedevice topography taken along line 1 e-1 f of FIG. 1E and fabricated byan exemplary method in accordance with an embodiment of the presentdisclosure.

As shown in FIGS. 1E-1F, in some embodiments, each respective element inthe corresponding plurality of elements comprises a corresponding cavityin a plurality of cavities engraved into the substrate in a directionsubstantially orthogonal to the native surface of the substrate. In someembodiments, a second graphene stack is formed on the native surface ofthe substrate.

FIG. 1G illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising a substantially radial arrangementof rows of elements formed as trenches engraved into a substrate withone or more graphene stacks on the native surface of the substrate andone or more graphene stacks in the trench surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure. FIG. 1H illustrates a cross-sectional view of theexemplary graphene device topography taken along line 1 g-1 h of FIG. 1Gand fabricated by an exemplary method in accordance with an embodimentof the present disclosure. As shown in FIGS. 1G-1H, in some embodiments,each respective element in the corresponding plurality of elementscomprises a corresponding cavity in a plurality of cavities engravedinto the substrate in a direction substantially orthogonal to the nativesurface of the substrate.

In some embodiments, the corresponding cavity is bounded by first endproximal to the native surface of the substrate and a second end distalto the native surface of the substrate; and a first graphene stack inthe at least one graphene stack is formed at the second end of therespective cavity.

In some embodiments, a second graphene stack is formed on the nativesurface of the substrate. In some embodiments, the first graphene stackhas a first set of properties and the second graphene stack has a secondset of properties, and the first set of properties is distinct from thesecond set of properties.

In some embodiments, the first graphene stack comprises M graphenelayers and the second graphene stack comprises P graphene layers, whereM and P are different positive integers greater than 1. In someembodiments, the plurality of rows includes a first row, a second rowadjacent to the first row, and a third row adjacent to the second row; areference axis of the first row is spaced at a first distance from areference axis of the second row; the reference axis of the second rowis spaced at a second distance from a reference axis of the third row;and the first distance is different than the second distance, therebyresulting in a non-linear array of elements.

In some embodiments, the first and second distance correspond to a firstand second predefined angular separation of values between 5° and 85°.

In some embodiments, the second dimension is non-orthogonal relative tothe first dimension. In some embodiments, the second dimension is at afixed predefined angle relative to the first dimension, therebyresulting in a substantially radial arrangement of the plurality ofrows. In some embodiments, the plurality of rows is configured along asubstantially radial arrangement and the second average spatialseparation is an average measure of an angular separation betweenconsecutive rows in the plurality of rows.

FIG. 1I illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed asprojections from a substrate with one or more graphene stacks on thebottom surfaces of the elements proximal to the native surface of thesubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure. FIG. 1J illustrates across-sectional view of the exemplary graphene device topography takenalong line 1 i-1 j of FIG. 1I and fabricated by an exemplary method inaccordance with an embodiment of the present disclosure. In someembodiments, each respective element in the corresponding plurality ofelements comprises a projection in a direction substantially orthogonalto the surface of the substrate. In some embodiments, each respectiveelement in the corresponding plurality of elements is a rib, a mesa, apillar, or any combination thereof. In some embodiments, the projectionincludes a first surface proximal to the surface of the substrate and asecond surface distal to the surface of the substrate; and a graphenestack in the at least one graphene stacks is formed on the first surfaceof the respective projection.

In some embodiments, a secondary layer is formed on the lateral surfaceof the projection; and the secondary layer comprises a secondarymaterial. In some embodiments, the secondary layer is a passivationlayer and the secondary material is a transmissive material. In someembodiments, the secondary layer comprises a waveguide guide tuned to apredetermined wavelength. In some embodiments, the secondary materialcomprises silicon nitride, silicon dioxide, titanium dioxide, tantalumoxide, hafnium oxide, or any combination thereof. In some embodiments,the secondary layer is an electrically isolating layer and the secondarymaterial is a dielectric. In some embodiments, the secondary materialcomprises a transparent conductive oxide. In some embodiments, thesecondary material comprises an electrically conductive material and thesecondary layer is an interconnect lead to electrically connect a firstconductive material with a second conductive material. In someembodiments, the N-dimensional array of elements forms a component of aDNA sequencing microcavity, a DNA sequencing nanocavity, a microfluidicchannel, a condenser lens, a fish eye lens, a diffraction area foroptical absorption, or any combination thereof. In some embodiments, thesecondary layer forms a component of the DNA sequencing microcavity, theDNA sequencing nanocavity, the microfluidic channel, the condenser lens,the fish eye lens, or the diffraction area for optical absorption.

FIG. 1K illustrates a top view of an exemplary graphene devicetopography (e.g., a plurality of elements along a row of an array formedas trenches engraved into a substrate with one or more graphene stackson the side walls of the engraved trenches) fabricated by exemplarymethods in accordance with an embodiment of the present disclosure. FIG.1L illustrates a cross-sectional view of the exemplary graphene devicetopography taken along line 1 k-1 l of FIG. 1K and fabricated by anexemplary method in accordance with an embodiment of the presentdisclosure. In some embodiments, each respective element in thecorresponding plurality of elements comprises a corresponding cavity ina plurality of cavities in the substrate; each respective cavity in theplurality of substrates includes (i) a corresponding trench surfaceformed on a surface of the cavity distal from the native surface of thesubstrate and (ii) a corresponding side wall formed between the nativesurface of the substrate and the corresponding trench surface of thecavity. In some embodiments, the corresponding side wall is formed at apredetermined angle to the corresponding trench surface of the cavity.In some embodiments, the predetermined angle has a value between 5° and85° measured with reference to the corresponding trench surface of thecavity. In some embodiments, a graphene stack the at least one graphenestack is formed on the corresponding side wall of the respective cavity.

FIGS. 2A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of non-uniformlyspaced elements formed as projections from a substrate with one or moregraphene stacks on the top surfaces of the elements distal from thenative surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure. FIG. 2Billustrates a cross-sectional view of the exemplary graphene devicetopography taken along line 2 a-2 b of FIG. 2A and fabricated by anexemplary method in accordance with an embodiment of the presentdisclosure.

FIG. 2C illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe trench surfaces of the trenches and a secondary layer formed overthe one or more graphene stacks of one or more of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure. FIG. 2D illustrates a cross-sectional view of theexemplary graphene device topography taken along line 2 c-2 d of FIG. 2Cand fabricated by an exemplary method in accordance with an embodimentof the present disclosure. As shown in FIGS. 2C-2D, in some embodiments,each respective element in the corresponding plurality of elementscomprises a corresponding cavity in a plurality of cavities engravedinto the substrate in a direction substantially orthogonal to the nativesurface of the substrate. In some embodiments, the corresponding cavityis bounded by first end proximal to the native surface of the substrateand a second end distal to the native surface of the substrate; and afirst graphene stack in the at least one graphene stack is formed at thesecond end of the respective cavity. In some embodiments, a secondarylayer is formed over the first graphene stack; and the secondary layercomprises a secondary material. In some embodiments, the secondary layeris a passivation layer and the respective secondary material is atransmissive material. In some embodiments, the secondary layercomprises a waveguide guide tuned to a respective wavelength. In someembodiments, the secondary material comprises silicon nitride, silicondioxide, titanium dioxide, tantalum oxide, hafnium oxide, or anycombination thereof. In some embodiments, the secondary layer is anelectrically isolating layer and the secondary material is a dielectric.In some embodiments, the secondary material comprises a transparentconductive oxide. In some embodiments, the secondary material comprisesan electrically conductive material and the secondary layer is aninterconnect lead to electrically connect a first conductive materialwith a second conductive material. In some embodiments, theN-dimensional array of elements forms a component of a DNA sequencingmicrocavity, a DNA sequencing nanocavity, a microfluidic channel, acondensor lens, a fish eye lens, a diffraction area for opticalabsorption, or any combination thereof. In some embodiments, therespective secondary layer forms a component of the DNA sequencingmicrocavity, the DNA sequencing nanocavity, the microfluidic channel,the condensor lens, the fish eye lens, the diffraction area for opticalabsorption, or any combination thereof.

FIG. 2E illustrates a top view of an exemplary graphene devicetopography (e.g., an array comprising rows of elements formed astrenches engraved into a substrate with one or more graphene stacks onthe native surface of the substrate) fabricated by exemplary methods inaccordance with an embodiment of the present disclosure. FIG. 2Fillustrates a cross-sectional view of the exemplary graphene devicetopography taken along line 2 e-2 f of FIG. 2E and fabricated by anexemplary method in accordance with an embodiment of the presentdisclosure.

FIGS. 3A-3C illustrate top views of exemplary graphene devicetopographies fabricated by exemplary methods in accordance withembodiments of the present disclosure. In some embodiments, the firstelement in the plurality of elements 104 is a rib. In some embodiments,the rib has a length and a width, where the length is at least two timesthe width. In some embodiments, the rib has a length and a width, wherethe length is between two times and five times the width. For example,the elements 104 (e.g., element 104-1) shown in FIG. 3A are ribs havinga length and a width, where the length is at least two times the width.In various embodiments, a width of a rib is between 1 nm and 10 nm,between 10 nm and 20 nm, between 20 nm and 30 nm, between 30 nm and 40nm, between 50 nm and 100 nm, or between 100 nm and 500 nm. In someembodiments, each rib in a plurality of ribs has a width that is on theorder of nanometers in width and this width does not deviate from thewidth of any other rib in the plurality of ribs by more than 0.1 nm, bymore than 0.2 nm, by more than 0.3 nm, by more than 0.4 nm, by more than0.5 nm, by more than 0.6 nm, by more than 0.7 nm, by more than 0.8 nm,by more than 0.9 nm, by more than 1 nm, by more than 2 nm, by more than3 nm, by more than 4 nm, or by more than 5 nm.

In some embodiments, a first element in the plurality of elements 104 isa mesa. In some embodiments, a mesa is an island isolated from otherfeatures on the substrate or a plateau on the substrate. In someembodiments, a mesa has at least one dimension (e.g., width or length)that is relatively large and thus can be used as a basis for furtherprocessing of more complex structures. In some embodiments, a mesa has atopographical height feature, providing a capability for verticalisolation and/or size for desired functionality. In some embodiments,the at least one dimension of the mesa is between 10 nm and 100 nm,between 100 nm and 1 μm, or between 1 μm and 10 μm. In some embodiments,the largest dimension of the mesa (e.g., width or length) is on theorder of nanometers and does not deviate from the largest dimension ofany other mesa in a plurality of mesas by more than 0.1 nm, by more than0.2 nm, by more than 0.3 nm, by more than 0.4 nm, by more than 0.5 nm,by more than 0.6 nm, by more than 0.7 nm, by more than 0.8 nm, by morethan 0.9 nm, by more than 1 nm, by more than 2 nm, by more than 3 nm, bymore than 4 nm, or by more than 5 nm.

In some embodiments, the first element in the plurality of elements 104is a pillar. In some embodiments, the top surface of the pillar (e.g.,pillar 104-2, FIG. 3B) is substantially circular. In some embodiments,the top surface of the pillar (e.g., pillar 104-6, FIG. 3C) issubstantially ovoid. In some embodiments, the top surface of the pillar(e.g., pillar 104-3, FIG. 3C) on is substantially polygonal. In someembodiments, the top surface of the pillar has an arcuate edge. In someembodiments, the elements 104 have holes within them.

FIG. 3D illustrates a cross-sectional view of exemplary graphene devicetopography (e.g., a row of elements with one or more graphene stackscovered with one or more secondary materials formed as projections on asubstrate) fabricated by exemplary methods in accordance with anembodiment of the present disclosure.

FIG. 4A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising rows of non-uniformlyspaced elements engraved as one or more layers of trenches into thesubstrate with one or more graphene stacks on the trench surfaces of theelements) fabricated by exemplary methods in accordance with anembodiment of the present disclosure. FIG. 4B illustrates across-sectional view of the exemplary graphene device topography takenalong line 4 a-4 b of FIG. 4A and fabricated by an exemplary method inaccordance with an embodiment of the present disclosure. In someembodiments, a first element in the corresponding plurality of elementsconsists of a single continuous cavity comprising a respective side walland a respective trench surface (e.g., elements 104-1 and 104-3, FIG.4B). In some embodiments, a graphene stack (e.g., 208-1, 208-3) in theat least one graphene stack is formed on the respective trench surfaceof the single continuous cavity. In some embodiments, a graphene stackin the at least one graphene stack is formed on the respective side wallof the single continuous cavity.

As shown in FIG. 4B, in some embodiments, a first element in thecorresponding plurality of elements includes a first sub-element (e.g.,104-2-a) and a second sub-element (e.g., 104-2-b). The first sub-elementcomprises an elemental cavity in the substrate in a directionsubstantially orthogonal to the surface of the substrate. The firstsub-element includes a respective elemental trench surface distal to thesurface of the substrate and a respective elemental side wall separatingthe surface of the substrate from the respective elemental trenchsurface. The second sub-element comprises a plurality of sub-elementalcavities (e.g., 140-1, 140-2, 140-3) formed along a direction parallelto a plane formed by the native surface of the substrate.

In some embodiments, each respective sub-elemental cavity in theplurality of sub-elemental cavities of the second sub-element is stackedcontiguous to the first sub-element along the direction substantiallyorthogonal to the native surface of the substrate and perforates theelemental trench surface of the first sub-element. Each respectivesub-elemental cavity in the plurality of sub-elemental cavities of thesecond sub-element comprises a respective sub-elemental side wall and arespective sub-elemental trench surface. In some embodiments, a graphenestack in the at least one graphene stack is segmentedly formed on therespective sub-elemental trench surface (e.g., 240-1, 240-2, 240-3) ofeach respective sub-elemental cavity.

In some embodiments, a secondary layer is formed over the graphene stackand the secondary layer comprises a secondary material. The secondarymaterial is optically transparent to a predefined range of wavelengthsof electromagnetic radiation incident on the N-dimensional array. Insome embodiments, the secondary material substantially fills eachrespective sub-elemental cavity in the plurality of sub-elementalcavities of the second sub-element. In some embodiments, the secondarylayer is a passivation layer and the respective secondary material is atransmissive material. In some embodiments, the secondary layercomprises a waveguide guide tuned to a respective wavelength. In someembodiments, the secondary material comprises silicon nitride, silicondioxide, titanium dioxide, tantalum oxide, hafnium oxide, or anycombination thereof. In some embodiments, the secondary layer is anelectrically isolating layer and the respective secondary material is adielectric. In some embodiments, the secondary material comprises atransparent conductive oxide. In some embodiments, the secondarymaterial comprises an electrically conductive material and therespective layer is an interconnect lead to electrically connect a firstconductive material with a second conductive material. In someembodiments, the N-dimensional array of elements forms a component of aDNA sequencing microcavity, a DNA sequencing nanocavity, a condensorlens, a flys eye lens, a diffraction area for optical absorption, or anycombination thereof. In some embodiments, the respective secondary layerforms the component of the DNA sequencing microcavity, the DNAsequencing nanocavity, the condensor lens, the flys eye lens, or thediffraction area for optical absorption.

In some embodiments, a graphene stack in the at least one graphene stackis segmentedly formed on the respective sub-elemental side wall of eachrespective sub-elemental cavity.

FIGS. 5A illustrates a top view of an exemplary graphene devicetopography (e.g., a non-linear array comprising hexagonal sub-arrayselements formed as projections on the substrate with one or moregraphene stacks on respective top or bottom surfaces of the elements)fabricated by exemplary methods in accordance with an embodiment of thepresent disclosure.

FIG. 5B and 5C illustrate cross-sectional views of the exemplarygraphene device topography taken along lines 5 b-5 b′ and 5 c-5 c′ ofFIG. 5A and fabricated by an exemplary method in accordance with anembodiment of the present disclosure.

As shown in FIG. 5A, in some embodiments, the N-dimensional array ofelements comprises a plurality of spatially-separated sub-arrays ofelements including a first element-sub-array (e.g., 502-a) and a secondelement sub-array (e.g., 502-b); the first element-sub-array comprises Melements (e.g., 7 elements) arranged in M1 rows (e.g., 3 rows), where Mis a positive integer of two or greater. The second element sub-arraycomprises P elements (e.g., 7 elements) arranged in P1 rows (e.g., 3rows), where P is a positive integer of two or greater. Each respectiveelement in the M elements in the first element-sub-array and eachrespective element in the P elements in the second element sub-arraycomprises at least one corresponding graphene stack (e.g., 504-a, 504-b,504-c, 504-d, and 504-e, as shown in FIGS. 5B, 5C). The set ofcharacteristic electromagnetic interference properties is associatedwith an interaction of a graphene stack in the M elements and a graphenestack in the P elements with the incident electromagnetic radiation.

As shown in FIG. 5A, in some embodiments, each respective element in thecorresponding plurality of elements (e.g., 104-1, 104-2, 104-3, and thelike) is a rib comprising a foundation material; a ratio of an averagelength of the rib to an average width of the rib has a value between 2and 10 and a graphene stack in the at least one graphene stack is formedover at least a portion of the foundation material (e.g., 504-a, 504-b,504-c, 504-d, and 504-e, as shown in FIGS. 5B, 5C).

In some embodiments, the corresponding plurality of elements includes afirst element characterized by a first length and a first width and asecond element characterized by a second length and a second width areference position of the first element is spaced a first distance froma reference position of the second element; and a ratio of the firstdistance to the first length has a value between 1.5 and 5.

In some embodiments, the first element comprises a first foundationmaterial and the second element comprises a second foundation material,distinct from the first foundation material; the first element comprisesa first graphene stack covering at least a portion of the firstrespective foundation material, the first graphene stack characterizedby a first set of properties based on properties of the first foundationmaterial; the second element comprises a second graphene stack coveringat least a portion of the second foundation material, the secondgraphene stack characterized by a second set of properties based onproperties of the second foundation material; and the first set ofproperties of the first graphene stack is distinct from the second setof properties of the second graphene stack.

In some embodiments, the set of characteristic electromagneticinterference properties is associated with an interaction of therespective graphene stack formed over at least the portion of therespective foundation material with the incident electromagneticradiation.

FIG. 6 illustrates a cross-sectional view of exemplary graphene devicetopography (e.g., quantum dots covered with one or more graphene stacks)fabricated by an exemplary method in accordance with an embodiment ofthe present disclosure.

In some embodiments, each respective element in the correspondingplurality of elements is a corresponding quantum dot (e.g., 104-1,104-2, 104-3) formed on the native surface of the substrate. The quantumdot comprises (i) a metal-based graphene initiating material (e.g.,602-1, 602-2, 602-3) formed on the native surface of the substrate(e.g., substrate 102) and (ii) the at least one graphene stack (e.g.,604-1, 604-2, 604-3) formed over at least a portion of the metal-basedgraphene initiating material. In some embodiments, the set ofcharacteristic electromagnetic interference properties is associatedwith an interaction of the at least one graphene stack formed over themetal-based graphene initiating material with the incidentelectromagnetic radiation.

FIGS. 7A-7H illustrate cross-sectional views of graphene devicetopography (e.g., including a plurality of contiguous cavities engravedinto the substrate) fabricated by an exemplary method in accordance withan embodiment of the present disclosure.

As shown in FIGS. 7A-7H, in some embodiments, each respective element inthe corresponding plurality of elements comprises a plurality ofcontiguous cavities in the substrate in a direction substantiallyorthogonal to the native lateral surface of the substrate. In suchembodiments, the plurality of contiguous cavities is stackedcontiguously along the direction substantially orthogonal to the surfaceof the substrate. In some embodiments, the plurality of contiguouscavities includes a first cavity (e.g., 704, FIGS. 7A-7C) comprising afirst reference axis (e.g., axis 730) orthogonal to the native surfaceof the substrate (e.g., substrate 102) and a second cavity (e.g., 720,FIGS. 7A-7C) comprising a second native axis orthogonal to the surfaceof the substrate. The first cavity has a first height defined along thefirst reference axis. The second cavity has a second height definedalong the second reference axis.

In some embodiments, each cavity in the plurality of contiguous cavitiesincludes respective side walls and respective trench surfaces and agraphene stack in the at least one graphene stack is segmentedly formedon each of the respective side walls of each cavity in the plurality ofcontiguous cavities.

In some embodiments, each cavity in the plurality of contiguous cavitiesincludes respective side walls and respective trench surfaces and agraphene stack in the at least one graphene stack is formed on one ormore of the respective trench surfaces of each cavity in the pluralityof contiguous cavities.

In some embodiments, the set of characteristic electromagneticinterference properties comprises a first respective inter-cavityresonance, occurring at a first respective wavelength of incidentelectromagnetic radiation, determined in accordance with the firstheight of the first cavity and a second respective wavelength ofincident electromagnetic radiation, determined in accordance with thesecond height of the second cavity.

In some embodiments, each respective cavity in the plurality ofcontiguous cavities includes a corresponding side wall and acorresponding trench surface. In some embodiments, the correspondingside wall is formed at a predetermined angle to the respective trenchsurface of each cavity, thereby resulting in a variable lateral width ofeach cavity between a first width measured proximal to the surface ofthe substrate and a second width measured proximal to the respectivetrench surface. In some embodiments, the predetermined angle has a valuebetween 5° and 85° measured relative to the respective trench surface.In some embodiments, a predetermined range of plasmonic resonancewavelengths of incident electromagnetic radiation is defined inaccordance with the variable lateral width of each cavity based on thepredetermined angle.

In some embodiments, each respective cavity in the plurality of cavitiesis characterized by an average spatial height along a third dimensionmeasured relative to the native surface of the substrate, therebyresulting in a third periodicity in lateral spacing along the thirddimension. In some embodiments, the third dimension is substantiallyorthogonal to the native lateral surface of the substrate. In someembodiments, the set of characteristic electromagnetic interferenceproperties comprises a third plasmonic resonance effect, occurring at athird resonance wavelength of incident electromagnetic radiation,characterized by the third periodicity in lateral spacing along thethird dimension and determined in accordance with the average spatialheight along the third dimension.

As shown in FIG. 7B, in some embodiments, a graphene stack in the atleast one graphene stack is segmentedly formed on each respective trenchsurface of each cavity in the plurality of contiguous cavities.

As shown in FIG. 7C, in some embodiments, a graphene stack in the atleast one graphene stack is segmentedly formed on each respective sidewall of each cavity in the plurality of contiguous cavities.

As shown in FIG. 7D, in some embodiments, a graphene stack (e.g., 740-1,740-2, 740-3) in the at least one graphene stack is segmentedly formedon at least a respective portion of one or more respective surfaces(e.g., at least a portion of a side wall and/or a portion of arespective trench surface; for example, as defined by a mask) one ormore cavities in the plurality of contiguous cavities.

As shown in FIG. 7E, in some embodiments, the substrate comprises afirst substrate layer (e.g., 750-1), a second substrate layer (e.g.,750-2), and a third substrate layer (e.g., 750-3), with the firstsubstrate layer defining the native surface of the substrate andoverlaying the second substrate layer, the second substrate layeroverlaying the third layer. In such embodiments, the first cavity is inthe first substrate layer and the second cavity extends to at least aportion of the second substrate layer.

In some embodiments, the first substrate layer substantially comprises afirst substrate material, the second substrate layer substantiallycomprises a second substrate material, and the third substrate layersubstantially comprises a third substrate material.

In some embodiments, the first, second and third substrate materials arerespectively different from each other. In some embodiments, the firstsubstrate material comprises a dielectric material, a conductivematerial, a semiconductor material, a doped semiconductor material, atransmissive material, an absorptive material, a transparent conductiveoxide, a metal oxide, a metal nitride, a metal, or a combinationthereof. In some embodiments, the second substrate material comprises adielectric material, a conductive material, a semiconductor material, adoped semiconductor material, a transmissive material, an absorptivematerial, a transparent conductive oxide, a metal oxide, a metalnitride, a metal, or a combination thereof. In some embodiments, thethird substrate material comprises a dielectric material, a conductivematerial, a semiconductor material, a doped semiconductor material, atransmissive material, an absorptive material, a transparent conductiveoxide, a metal oxide, a metal nitride, a metal, or a combinationthereof.

In some embodiments, the first substrate material comprises silicon,gallium arsenide, germanium, silicon dioxide, titanium dioxide, siliconnitride, hafnium oxide, tantalum oxide, zinc oxide, aluminum oxide,silicon carbide, silicon nitride, titanium nitride, tantalum nitride,neoceram, barosilicate glass, soda lime glass, lead glass, doped silicondioxide, doped n-type silicon, doped p-type silicon, aluminosilicateglass, dichroic glass, semiconductor glass, glass ceramic, silicate,fused silica, quartz, chalcogenide glass, sulphide glass, poly methylmethacrylate, polyethylene terephthalate, polyvinyl alcohol, celluloseacetate, boron nitride, molybdenum sulfide, zinc, copper, nickel, iron,platinum, gold, palladium, ruthenium, vanadium, hafnium, cadmium,tungsten, aluminum, titanium, cadmium, silver, or a combination thereof.

In some embodiments, the second substrate material comprises silicon,gallium arsenide, germanium, silicon dioxide, titanium dioxide, siliconnitride, hafnium oxide, tantalum oxide, zinc oxide, aluminum oxide,silicon carbide, silicon nitride, titanium nitride, tantalum nitride,neoceram, barosilicate glass, soda lime glass, lead glass, doped silicondioxide, doped n-type silicon, doped p-type silicon, aluminosilicateglass, dichroic glass, semiconductor glass, glass ceramic, silicate,fused silica, quartz, chalcogenide glass, sulphide glass, poly methylmethacrylate, polyethylene terephthalate, polyvinyl alcohol, celluloseacetate, boron nitride, molybdenum sulfide, zinc, copper, nickel, iron,platinum, gold, palladium, ruthenium, vanadium, hafnium, cadmium,tungsten, aluminum, titanium, cadmium, silver, or a combination thereof.

In some embodiments, the third substrate material comprises silicon,gallium arsenide, germanium, silicon dioxide, titanium dioxide, siliconnitride, hafnium oxide, tantalum oxide, zinc oxide, aluminum oxide,silicon carbide, silicon nitride, titanium nitride, tantalum nitride,neoceram, barosilicate glass, soda lime glass, lead glass, doped silicondioxide, doped n-type silicon, doped p-type silicon, aluminosilicateglass, dichroic glass, semiconductor glass, glass ceramic, silicate,fused silica, quartz, chalcogenide glass, sulphide glass, poly methylmethacrylate, polyethylene terephthalate, polyvinyl alcohol, celluloseacetate, boron nitride, molybdenum sulfide, zinc, copper, nickel, iron,platinum, gold, palladium, ruthenium, vanadium, hafnium, cadmium,tungsten, aluminum, titanium, cadmium, silver, or a combination thereof.

As shown in FIG. 7F, a graphene stack in the at least one graphene stackis a continuous graphene stack (e.g., 760-1) formed on a continuoussurface formed by each respective side wall and each respective trenchsurface of each cavity in the plurality of contiguous cavities.

In some embodiments, each contiguous cavity in the plurality ofcontiguous cavities has a common central axis (e.g., axis 730, FIGS.7A-7F) orthogonal to the native surface of the substrate.

In some embodiments, the plurality of contiguous cavities includes afirst cavity comprising a first reference axis orthogonal to the nativesurface of the substrate and a second cavity comprising a secondreference axis orthogonal to the native surface of the substrate; andthe first reference axis is distinct from and parallel to the secondreference axis.

FIGS. 8A, 8C and 8E illustrate top views of an exemplary graphene devicetopography (e.g., elements along a row of an array formed as projectionswith shapes corresponding to frustums of pyramids on the substrate withone or more graphene stacks on respective top surfaces, side walls, anda combination thereof) fabricated by exemplary methods in accordancewith an embodiment of the present disclosure.

FIGS. 8B, 8D and 8F illustrate cross-sectional views of the exemplarygraphene device topography taken along lines 8 a-8 b, 8 c-8 d and 8 e-8f of FIGS. 8A, 8C and 8E and fabricated by an exemplary method inaccordance with an embodiment of the present disclosure.

As shown in FIGS. 8A, 8C and 8E each respective element (e.g., 104-1,104-2, 104-3) in the corresponding plurality of elements is a frustum ofa pyramid. A base of the frustum is affixed to the surface of thesubstrate (e.g., substrate 102), the base being characterized by a firstsurface area and a first perimeter defining a first n-gon with n edges(e.g., a rectangle with 4 sides as shown in FIGS. 8A, 8C and 8E), wheren in an integer of three or greater, and having a first averageedge-length. A central axis of the frustum is defined substantiallyorthogonal to the lateral surface of the substrate between therespective base of the frustum and a respective top surface of thefrustum.

A top surface of the frustum is distal to the native surface of thesubstrate, the top surface of the frustum being characterized by asecond surface area and a second perimeter defining a second n-gon withN edges having a second average edge-length, the second surface areabeing less than the first surface area and the second averageedge-length being less than the first average edge-length; and Nrespective faces of the frustum separating the respective base of thefrustum from the respective top surface of the frustum and bounded bythe N edges of the respective base of the frustum and the respective Nedges of the respective top surface of the frustum.

As shown in FIG. 8B, in some embodiments, a first graphene stack (e.g.,802-1, 802-2, 802-3) in the at least one graphene stack is formed overthe top surface of the frustum.

As shown in FIG. 8D, in some embodiments, the first graphene stack(e.g., 802-1, 802-2, 802-3) is formed over each of the N respectivefaces of the frustum.

As shown in FIG. 8F, in some embodiments, the first graphene stack(e.g., 802-1, 802-2, 802-3) is formed over the top surface of thefrustum and over each of the N respective faces of the frustum.

In some embodiments, the first graphene stack is formed over one or moreof: the top surface and the N respective faces of the frustum; a secondgraphene stack is formed over the surface of the substrate; the firstgraphene layer has a first set of properties and the second graphenestack has a second set of properties; and the first set of properties isdistinct from the second set of properties.

In some embodiments, the first graphene stack is formed over one or moreof: the top surface and the N respective faces of the frustum; and theset of characteristic electromagnetic interference properties isassociated with an interaction of the first graphene stack with theincident electromagnetic radiation.

FIGS. 9A-9E illustrate a flow diagram representing exemplary processflow 900 for the fabrication of one or more graphene device topographyin accordance with an embodiment of the present disclosure. As shown inFIGS. 9A-9E, the fabrication process flow 900 includes patterning (902)a substrate. In particular, the substrate is patterned to form anN-dimensional array of elements on a lateral surface of a substrate. TheN-dimensional array includes a plurality of rows, each respective row inthe plurality of rows comprises a plurality of elements formed along afirst dimension characterized by an axis of the respective row, eachrespective element in the plurality of elements is separated from anadjacent element along the first dimension by a first average spatialseparation, thereby resulting in a first periodicity in lateral spacingalong the first dimension. Each respective row in the plurality of rowsis separated from an adjacent row along a second dimension by a secondaverage spatial separation, thereby resulting in a second periodicity inlateral spacing along the second dimension.

In some embodiments, the N-dimensional array is a linear array (e.g.,with uniform spacing between consecutive elements and/or betweenconsecutive rows) (904). In some embodiments, the N-dimensional array isa non-linear array (e.g., with non-uniform spacing between consecutiveelements and/or between consecutive rows) (906). In some embodiments,the N-dimensional array comprises a substantially radial arrangement ofthe plurality of rows (e.g., with a predefined angle of separationbetween consecutive rows of the plurality of rows) (908). In someembodiments, the plurality of elements is formed by engraving (e.g.,etching) a plurality of cavities into the substrate in a directionsubstantially orthogonal to the native lateral surface of the substrate(910). In some embodiments, the plurality of elements is formed bygrowing portions of substrate material or growing at least onerespective secondary material selectively at regions of the substrate orthe secondary material corresponding to the plurality of elements (912).

In some embodiments, the substrate substantially comprises a materialselected from the group consisting of: neoceram, barosilicate glass,germanium arsenide, a IV-V semiconductor material, a substantiallymetallic material, a high temperature glass, and a combination thereof(914). In some embodiments, the substrate substantially comprises SiO₂glass, soda lime glass, lead glass, doped SiO₂, aluminosilicate glass,borosilicate glass, dichroic glass, germanium/semiconductor glass, glassceramic, silicate/fused silica, quartz or chalcogenide/sulphide glass(916).

In some embodiments, each respective element in the plurality ofelements comprises a respective projection in a direction substantiallyorthogonal to the lateral surface of the substrate (918).

In some embodiments, each respective element in the plurality ofelements comprises a corresponding cavity in a plurality of cavitiesengraved into the substrate in a direction substantially orthogonal tothe native lateral surface of the substrate (920).

In some embodiments, each respective element in the plurality ofelements comprises a plurality of contiguous cavities in the substratein a direction substantially orthogonal to the native lateral surface ofthe substrate, and wherein the plurality of contiguous cavities isstacked contiguously along the direction substantially orthogonal to thelateral surface of the substrate (922).

In some embodiments, each respective element in the plurality ofelements is a quantum dot formed on the native lateral surface of thesubstrate. The quantum dot further includes a respective graphene stackformed over the quantum dot to cover at least a portion of a metal-basedgraphene initiating material on a respective surface of the respectivenodule distal from the lateral surface of the substrate (924).

In some embodiments, each respective element in the plurality ofelements is a rib comprising a respective foundation material and aratio of an average length of the rib to an average width of the rib hasa value between 2 and 10 (926).

In some embodiments, each respective element in the plurality ofelements is a frustum of a pyramid (928).

The method further includes depositing (930) a respective grapheneinitiating layer onto a respective surface of each element of theplurality of elements. In some embodiments, the depositing of therespective graphene initiating layer includes sputter-depositing therespective graphene initiating layer onto the respective surface of eachrespective element in the plurality of elements (932). In someembodiments, the respective graphene initiating layer substantiallycomprises a metal selected from the group consisting of: platinum, gold,palladium, ruthenium, aluminum, titanium, tungsten, cadmium, copper,nickel, nickel foam, and iron (934). In some embodiments, the respectivegraphene initiating layer substantially comprises a compound of carbon(936). In some embodiments, the respective graphene initiating layercomprises silicon (938).

The method further includes generating (942) a respective graphene stackon the respective surface of each element of the plurality of elementsusing the respective graphene initiating layer. In some embodiments, therespective graphene initiating layer substantially comprises a metal(see step 934), and generating the respective graphene stack using therespective graphene initiating layer comprises growing a carbon materialon the respective graphene initiating layer thereby forming thegraphite-based structure (944). In some embodiments the carbon materialis deposited on the respective graphene initiating layer (946). In someembodiments the deposited carbon material is heated thereby forming therespective graphene stack (948). In some embodiments, when therespective graphene initiating layer substantially comprises a compoundof carbon (see step 936), generating the respective graphene stack usingthe respective graphene initiating layer comprises heating therespective graphene initiating layer to vaporize an element other thancarbon from the compound of carbon (950). In some embodiments, when thecompound of carbon is silicon carbide (see step 938), generating therespective graphene stack using the respective graphene initiating layercomprises heating the silicon carbide to vaporize elemental silicon inthe silicon carbide (952).

In some embodiments (954), when the respective graphene initiating layercomprises silicon (see step 940), generating the respective graphenestack using the respective graphene initiating layer comprises: (i)depositing elemental carbon on or into the respective silicon grapheneinitiating layer (956), e.g., by doping the respective silicon grapheneinitiating layer by implantation of the elemental carbon into therespective silicon graphene initiating layer (958), and (ii) convertingthe respective silicon graphene initiating layer into silicon carbide byenabling a chemical reaction of the respective silicon grapheneinitiating layer with the deposited elemental carbon (960), e.g., byheating the silicon carbide to vaporize the silicon from the siliconcarbide by reverse epitaxy, thereby forming the respective graphenestack (962).

In some embodiments, each respective element in the plurality ofelements comprises at least one respective graphene stack, and theN-dimensional array exhibits a set of characteristic electromagneticinterference properties in response to electromagnetic radiationincident on the N-dimensional array (964). In some embodiments, the setof characteristic electromagnetic interference properties is associatedwith an interaction of the at least one respective graphene stack withthe incident electromagnetic radiation (966). In some embodiments, theset of characteristic electromagnetic interference properties comprisesa first plasmonic resonance, occurring at a first resonance wavelengthof incident electromagnetic radiation, characterized by the firstperiodicity in lateral spacing along the first dimension and determinedin accordance with the first average spatial separation along the firstdimension (968). In some embodiments, the set of characteristicelectromagnetic interference properties comprises a second plasmonicresonance, occurring at a second resonance wavelength of incidentelectromagnetic radiation, characterized by the second periodicity inlateral spacing along the second dimension and determined in accordancewith the second average spatial separation along the second dimension(970).

The foregoing description, for purpose of explanation, has beendescribed with reference to specific embodiments. However, theillustrative discussions above are not intended to be exhaustive or tolimit the invention to the precise forms disclosed. Many modificationsand variations are possible in view of the above teachings. Theembodiments were chosen and described in order to best explain theprinciples of the invention and its practical applications, to therebyenable others skilled in the art to best utilize the invention andvarious embodiments with various modifications as are suited to theparticular use contemplated.

1-77. (canceled)
 78. An integrated graphene-based structure comprisingan N-dimensional array of elements formed on a surface of a substrate,wherein: (i) the N-dimensional array of elements includes a plurality ofrows, (ii) each respective row in the plurality of rows comprises acorresponding plurality of elements formed along a first dimension, thefirst dimension characterized by an axis of the respective row, eachelement in the corresponding plurality of elements (a) comprising atleast one graphene stack and (b) separated from an adjacent elementalong the first dimension by a first average spatial separation therebyresulting in a first periodicity in lateral spacing along the firstdimension, (iii) each respective row in the plurality of rows isseparated from an adjacent row along a second dimension by a secondaverage spatial separation, thereby resulting in a second periodicity inlateral spacing along the second dimension, and (iv) the N-dimensionalarray exhibits a set of characteristic electromagnetic interferenceproperties in response to electromagnetic radiation incident on theN-dimensional array, wherein: each respective element in thecorresponding plurality of elements comprises a corresponding cavity ina plurality of cavities in the substrate; each respective cavity in theplurality of substrates includes (i) a corresponding trench surfaceformed on a surface of the cavity distal from the native surface of thesubstrate and (ii) a corresponding side wall formed between the nativesurface of the substrate and the corresponding trench surface of thecavity.
 79. The structure of claim 78, wherein the corresponding sidewall is formed at a predetermined angle to the corresponding trenchsurface of the cavity.
 80. The structure of claim 79, wherein thepredetermined angle has a value between 5° and 85° measured withreference to the corresponding trench surface of the cavity.
 81. Thestructure of claim 78, wherein a graphene stack the at least onegraphene stack is formed on the corresponding side wall of therespective cavity.
 82. An integrated graphene-based structure comprisingan N-dimensional array of elements formed on a surface of a substrate,wherein: (i) the N-dimensional array of elements includes a plurality ofrows, (ii) each respective row in the plurality of rows comprises acorresponding plurality of elements formed along a first dimension, thefirst dimension characterized by an axis of the respective row, eachelement in the corresponding plurality of elements (a) comprising atleast one graphene stack and (b) separated from an adjacent elementalong the first dimension by a first average spatial separation therebyresulting in a first periodicity in lateral spacing along the firstdimension, (iii) each respective row in the plurality of rows isseparated from an adjacent row along a second dimension by a secondaverage spatial separation, thereby resulting in a second periodicity inlateral spacing along the second dimension, and (iv) the N-dimensionalarray exhibits a set of characteristic electromagnetic interferenceproperties in response to electromagnetic radiation incident on theN-dimensional array, wherein: the plurality of elements includes a firstelement comprising a first graphene stack and a second elementcomprising a second graphene stack; the first element is characterizedby a first width and the second element is characterized by a secondwidth, wherein the first width is different from the second width; and areference position of the first element is separated from a referenceposition of the second element by a first spatial separation.
 83. Thestructure of claim 82, wherein the set of characteristic electromagneticinterference properties comprises: a first elemental resonance,occurring at a first elemental wavelength of incident electromagneticradiation, determined in accordance with the first lateral width of thefirst element and associated with an interaction of the first graphenestack with the incident electromagnetic radiation; a second elementalresonance, occurring at a second elemental wavelength of incidentelectromagnetic radiation, determined in accordance with the secondlateral width of the second element and associated with an interactionof the second graphene stack with the incident electromagneticradiation; and a respective inter-elemental resonance, occurring at athird elemental wavelength of incident electromagnetic radiation,determined in accordance with the first spatial separation.
 84. Thestructure of claim 83, wherein the first element comprises a firstcavity in the substrate in a direction substantially orthogonal to thesurface of the substrate and the second element comprises a secondcavity in the substrate in the direction substantially orthogonal to thesurface of the substrate.
 85. The structure of claim 84, wherein: thefirst cavity includes a first end proximal to the native surface of thesubstrate and a second end distal to the native surface of thesubstrate; and the first graphene stack is formed on the second end ofthe first cavity.
 86. The structure of claim 85, wherein: a secondarylayer is formed over the first graphene stack; and the secondary layercomprises a secondary material.
 87. The structure of claim 86, whereinthe secondary layer is a passivation layer and the respective secondarymaterial is a transmissive material.
 88. The structure of claim 86,wherein the secondary layer comprises a waveguide guide tuned to arespective wavelength.
 89. The structure of claim 88, wherein thesecondary material comprises silicon nitride, silicon dioxide, titaniumdioxide, tantalum oxide, hafnium oxide, or any combination thereof. 90.The structure of claim 86, wherein the secondary layer is anelectrically isolating layer and the secondary material is a dielectric.91. The structure of claim 86, wherein the secondary material comprisesa transparent conductive oxide.
 92. The structure of claim 86, whereinthe secondary material comprises an electrically conductive material andthe secondary layer is an interconnect lead to electrically connect afirst conductive material with a second conductive material.
 93. Thestructure of claim 86, wherein the N-dimensional array of elements formsa component of a DNA sequencing microcavity, a DNA sequencingnanocavity, a condensor lens, a flys eye lens, a diffraction area foroptical absorption, or any combination thereof.
 94. The structure ofclaim 93, wherein the secondary layer forms a component of the DNAsequencing microcavity, the DNA sequencing nanocavity, the condensorlens, the flys eye lens, or the diffraction area for optical absorption.95. The structure of claim 85, wherein a second graphene stack is formedon the lateral surface of the substrate.
 96. The structure of claim 95,wherein: the first graphene stack has a first set of properties and thesecond graphene stack has a second set of properties; and the first setof properties is distinct from the second set of properties.
 97. Thestructure of claim 84, wherein: the first element is adjacent to thesecond element; the plurality of elements further includes a thirdelement adjacent to the second element, wherein the third elementcomprises a third cavity characterized by a third lateral width, andwherein the third lateral width, the second lateral width, and the firstlateral width are distinct from each other; and the reference positionof the second element is separated from a reference position of thethird element by a second spatial separation.
 98. The structure of claim84, wherein the first cavity includes a first trench surface formed on asurface of the cavity distal from the native surface of the substrateand a first side wall is formed between the native surface of thesubstrate and the respective trench surface of the cavity.
 99. Thestructure of claim 98, wherein the first side wall is formed at apredetermined angle to the respective trench surface of the cavity. 100.The structure of claim 99, wherein the predetermined angle has a valuebetween 5° and 85° measured with reference to the respective trenchsurface of the cavity.
 101. The structure of claim 98, wherein the firstgraphene stack is formed on the first side wall of the first cavity.102-155. (canceled)